On the Hartley – Fourier sine generalized convolution

In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H₁,H₂) and the Fourier sine (F_s) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities:

We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.